When m gram of steam at `100^(@) C` is mixed with 200 gm of ice at `0^(@)C`. it results in water at `40^(@) C`. Find the value of m in gram . (given : Latent heat of fusion (`L_f`) = 80 cal/gm, Latent heat of vaporisation (`L_v`) = 540 cal/gm., specific heat of water (`C_w`)= 1 cal/gm/C)

Explanation

To solve the problem, we will apply the principle of calorimetry, which states that the heat gained by the ice will be equal to the heat lost by the steam.

### Step-by-Step Solution:

1. **Identify the Heat Gained by Ice:**
- The ice at 0°C will first absorb heat to melt into water at 0°C. The heat gained during this phase is given by:
\[ Q_1 = m_{\text{ice}} \cdot L_f \]
where \( m_{\text{ice}} = 200 \, \text{g} \) and \( L_f = 80 \, \text{cal/g} \).
- After melting, the water at 0°C will absorb heat to raise its temperature to 40°C. The heat gained during this phase is:
\[ Q_2 = m_{\text{water}} \cdot C_w \cdot \Delta T \]
where \( m_{\text{water}} = 200 \, \text{g} \), \( C_w = 1 \, \text{cal/g°C} \), and \( \Delta T = 40°C - 0°C = 40°C \).

2. **Calculate the Total Heat Gained by Ice:**
- The total heat gained by the ice is:
\[ Q_{\text{ice}} = Q_1 + Q_2 = (200 \cdot 80) + (200 \cdot 1 \cdot 40) \]
\[ Q_{\text{ice}} = 16000 + 8000 = 24000 \, \text{cal} \]

3. **Identify the Heat Lost by Steam:**
- The steam at 100°C will first condense into water at 100°C. The heat lost during this phase is:
\[ Q_3 = m_{\text{steam}} \cdot L_v \]
where \( L_v = 540 \, \text{cal/g} \).
- The water at 100°C will then cool down to 40°C. The heat lost during this phase is:
\[ Q_4 = m_{\text{steam}} \cdot C_w \cdot \Delta T \]
where \( \Delta T = 100°C - 40°C = 60°C \).

4. **Calculate the Total Heat Lost by Steam:**
- The total heat lost by the steam is:
\[ Q_{\text{steam}} = Q_3 + Q_4 = (m \cdot 540) + (m \cdot 1 \cdot 60) \]
\[ Q_{\text{steam}} = m \cdot (540 + 60) = m \cdot 600 \]

5. **Set Heat Gained Equal to Heat Lost:**
- According to the principle of calorimetry:
\[ Q_{\text{ice}} = Q_{\text{steam}} \]
\[ 24000 = m \cdot 600 \]

6. **Solve for m:**
- Rearranging gives:
\[ m = \frac{24000}{600} = 40 \, \text{g} \]

### Final Answer:
The value of \( m \) is **40 grams**.